Question: Find the $1314^{\text{th}}$ digit past the decimal point in the decimal expansion of $\dfrac{5}{14}$.
Solution: First, we find the repeating decimal expansion of 5/14: $$ \frac{5}{14} = \frac{5}{5} \cdot \frac{5}{14} = \frac{25}{70} = \frac{25}{7} \cdot \frac{1}{10} = (3.\overline{571428})(0.1) = 0.3\overline{571428}. $$The $1314^{\text{th}}$ digit after the decimal point is the $1313^{\text{th}}$ digit in the 6-digit repeating block 5-7-1-4-2-8.  Since $1313 \div 6$ leaves a remainder of 5, our answer is the $5^{\text{th}}$ digit in the 6-digit block, which is $\boxed{2}$.